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A281140
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Least k such that k is the product of n distinct primes and sigma(k) is an n-th power.
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2
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2, 22, 102, 510, 90510, 995610, 11616990, 130258590, 1483974030, 18404105922510, 428454465915630, 10195374973815570, 240871269907008510, 94467020965716904490370, 4580445736068712946096430, 7027383957579235221501981990, 419420669769073022876839238610, 24967450935148397377034326845390
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OFFSET
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1,1
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COMMENTS
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Freiberg (Theorem 1.2) shows that there are >> (n*x^(1/n))/(log x)^(n+2) such values of k up to x. He calls the set of such numbers B*(x;+1;n). In particular, a(n) exists for each n.
Corresponding values of sigma(k) are 3 = 3^1, 36 = 6^2, 216 = 6^3, 1296 = 6^4, 248832 = 12^5, 2985984 = 12^6, 12^7, 12^8, 12^9, 24^10, 24^11, 24^12, 24^13, 48^14, 48^15, 60^16, 60^17, 60^18, 60^19, 84^20, 84^21, 84^22, 84^23, ...
a(14) <= 94467020965716904490370. - Daniel Suteu, Mar 28 2021
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LINKS
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EXAMPLE
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a(3) = 102 because 102 = 2 * 3 * 17 and (2 + 1)*(3 + 1)*(17 + 1) = 6^3.
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PROG
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(PARI) a(n) = my(k=2); while(!issquarefree(k) || !ispower(sigma(k), n) || omega(k)!=n, k++); k \\ Felix Fröhlich, Jan 17 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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